The abstract Perron-Stieltjes integral in the Kurzweil-Henstock sense given via integral sums is used for defining convolutions of Banach space valued functions. Basic facts concerning integration are preseted, the properties of Stieltjes convolutions are studied and applied to obtain resolvents for renewal type Stieltjes convolution equations.

We present an operator-valued version of the conditionally free product of states and measures, which in the scalar case was studied by Bożejko, Leinert and Speicher. The related combinatorics and limit theorems are provided.

Two of James’ three quasi-reflexive spaces, as well as the James Tree, have the uniform $w^{*}$-Opial property.

We find the norm of the Hardy operator minus the identity acting on the cone of radially decreasing functions on minimal Lorentz spaces (restricted type estimates).

For $\left(P_k\right)$ being Rademacher, Fermion or q-Gaussian (-1 ≤ q ≤ 0) operators, we find the optimal constants $C_{2n}$, n∈ ℕ, in the inequality $\parallel {\sum }_{k=1}^N{A}_k\otimes P_k{\parallel }_{2n}\le {\left[C_{2n}\right]}^{1/2n}max{\parallel ({\sum }_{k=1}^{N}A{*}_k{A}_k}^{1/2}{\parallel }_{L_{2n}},\parallel ({\sum }_{k=1}^N{A}_{k}A{*}_k$1/2∥L2n$,$valid for all finite sequences of operators $\left(A_k\right)$ in the non-commutative $L_{2n}$ space related to a semifinite von Neumann algebra with trace. In particular, $C_{2n}=(2nr-1)!!$ for the Rademacher and Fermion sequences.

Sequences of cubature formulas with a joint countable set of nodes are studied. Each cubature formula under consideration has only a finite number of nonzero weights. We call a sequence of such kind a multicubature formula. For a given reflexive Banach space it is shown that there is a unique optimal multicubature formula and the sequence of the norm of optimal error functionals is monotonically decreasing to 0 as the number of the formula nodes tends to infinity.

Let T be a kernel operator with values in a rearrangement invariant Banach function space X on [0,∞) and defined over simple functions on [0,∞) of bounded support. We identify the optimal domain for T (still with values in X) in terms of interpolation spaces, under appropriate conditions on the kernel and the space X. The techniques used are based on the relation between linear operators and vector measures.

Refinements of the classical Sobolev inequality lead to optimal domain problems in a natural way. This is made precise in recent work of Edmunds, Kerman and Pick; the fundamental technique is to prove that the (generalized) Sobolev inequality is equivalent to the boundedness of an associated kernel operator on [0,1]. We make a detailed study of both the optimal domain, providing various characterizations of it, and of properties of the kernel operator when it is extended to act in its optimal domain....

We establish the embedding of the critical Sobolev-Lorentz-Zygmund space $H_{p,q,\lambda ₁,...,\lambda ₘ}^{n/p}\left(ℝⁿ\right)$ into the generalized Morrey space ${ℳ}_{\Phi ,r}\left(ℝⁿ\right)$ with an optimal Young function Φ. As an application, we obtain the almost Lipschitz continuity for functions in $H_{p,q,\lambda ₁,...,\lambda ₘ}^{n/p+1}\left(ℝⁿ\right)$. O’Neil’s inequality and its reverse play an essential role in the proofs of the main theorems.

We prove optimal embeddings of homogeneous Sobolev spaces built over function spaces in ℝⁿ with K-monotone and rearrangement invariant norm into other rearrangement invariant function spaces. The investigation is based on pointwise and integral estimates of the rearrangement or the oscillation of the rearrangement of f in terms of the rearrangement of the derivatives of f.

This paper is a survey of recent results on some problems of supervised learning in the setting formulated by Cucker and Smale. Supervised learning, or learning-from-examples, refers to a process that builds on the base of available data of inputs $x_i$ and outputs $y_i$, i = 1,...,m, a function that best represents the relation between the inputs x ∈ X and the corresponding outputs y ∈ Y. The goal is to find an estimator $f_z$ on the base of given data $z:=((x₁,y₁),...,(x_m,y_m))$ that approximates well the regression function $f_{\rho }$ of...

Dato un qualsiasi spazio invariante per riordinamenti $X\left(\mathrm{\Omega }\right)$ su un insieme aperto $\mathrm{\Omega }\subset {\mathbb{R}}^n$, si determina il più piccolo spazio invariante per riordinamenti $Y\left(\mathrm{\Omega }\right)$ con la proprietà che se $u:\mathrm{\Omega }\to {\mathbb{R}}^n$ è una applicazione che mantiene l'orientamento e ${\left|Du\right|}^n\in X\left(\mathrm{\Omega }\right)$, allora $detDu$ appartiene localmente a $Y\left(\mathrm{\Omega }\right)$.

We present optimal upper bounds for expectations of order statistics from i.i.d. samples with a common distribution function belonging to the restricted family of probability measures that either precede or follow a given one in the star ordering. The bounds for families with monotone failure density and rate on the average are specified. The results are obtained by projecting functions onto convex cones of Hilbert spaces.